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1 Inventory Management and Control

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2 Inventory Defined Inventory is the stock of any item or resource held to meet future demand and can include: raw materials, finished products, component parts, supplies, and work-in-process

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3 Inventory Process stage Demand Type Number & Value Other Raw Material WIP Finished Goods Independent Dependent A Items B Items C Items Maintenance Operating Inventory Classifications

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4 E(1 ) Independent vs. Dependent Demand B(4) E(2)D(1) C(2) E(3)B(1) A Independent Demand (Demand for the final end-product or demand not related to other items; demand created by external customers) Dependent Demand (Derived demand for component parts, subassemblies, raw materials, etc- used to produce final products ) Finished product Component parts Independent demand is uncertain Dependent demand is certain

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5 Inventory Models Independent demand – finished goods, items that are ready to be sold –E.g. a computer Dependent demand – components of finished products –E.g. parts that make up the computer

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6 Types of Inventories (1 of 2) Raw materials & purchased parts Partially completed goods called work in progress Finished-goods inventories (manufacturing firms) or merchandise (retail stores)

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7 Types of Inventories (2 of 2) Replacement parts, tools, & supplies Goods-in-transit to warehouses or customers

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8 The Material Flow Cycle (1 of 2)

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9 Run time: Job is at machine and being worked on Setup time: Job is at the work station, and the work station is being "setup." Queue time: Job is where it should be, but is not being processed because other work precedes it. Move time: The time a job spends in transit Wait time: When one process is finished, but the job is waiting to be moved to the next work area. The Material Flow Cycle (2 of 2) Wait Time Move Time Queue Time Setup Time Run Time Input Cycle Time Output

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10 Performance Measures Inventory turnover (the ratio of annual cost of goods sold to average inventory investment) Days of inventory on hand (expected number of days of sales that can be supplied from existing inventory)

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11 Functions of Inventory (1 of 2) 1.To decouple or separate various parts of the production process, ie. to maintain independence of operations 2.To meet unexpected demand & to provide high levels of customer service 3.To smooth production requirements by meeting seasonal or cyclical variations in demand 4.To protect against stock-outs

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12 Functions of Inventory (2 of 2) 5. To provide a safeguard for variation in raw material delivery time 6. To provide a stock of goods that will provide a selection for customers 7. To take advantage of economic purchase-order size 8. To take advantage of quantity discounts 9. To hedge against price increases

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13 Higher costs –Item cost (if purchased) –Holding (or carrying) cost Difficult to control Hides production problems May decrease flexibility Disadvantages of Inventory

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14 Inventory Costs Holding (or carrying) costs Costs for storage, handling, insurance, etc Setup (or production change) costs Costs to prepare a machine or process for manufacturing an order, eg. arranging specific equipment setups, etc Ordering costs (costs of replenishing inventory) Costs of placing an order and receiving goods Shortage costs Costs incurred when demand exceeds supply

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15 Holding (Carrying) Costs Obsolescence Insurance Extra staffing Interest Pilferage Damage Warehousing Etc.

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16 Inventory Holding Costs (Approximate Ranges) Category Housing costs (building rent, depreciation, operating cost, taxes, insurance) Material handling costs (equipment, lease or depreciation, power, operating cost) Labor cost from extra handling Investment costs (borrowing costs, taxes, and insurance on inventory) Pilferage, scrap, and obsolescence Overall carrying cost Cost as a % of Inventory Value 6% (3 - 10%) 3% (1 - 3.5%) 3% (3 - 5%) 11% (6 - 24%) 3% (2 - 5%) 26%

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17 Ordering Costs Supplies Forms Order processing Clerical support etc.

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18 Setup Costs Clean-up costs Re-tooling costs Adjustment costs etc.

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19 Shortage Costs Backordering cost Cost of lost sales

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20 Inventory Control System Defined An inventory system is the set of policies and controls that monitor levels of inventory and determine what levels should be maintained, when stock should be replenished and how large orders should be Answers questions as: When to order? How much to order?

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21 Objective of Inventory Control To achieve satisfactory levels of customer service while keeping inventory costs within reasonable bounds Improve the Level of customer service Reduce the Costs of ordering and carrying inventory

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22 A system to keep track of inventory A reliable forecast of demand Knowledge of lead times Reasonable estimates of Holding costs Ordering costs Shortage costs A classification system Requirements of an Effective Inventory Management

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23 Inventory Counting (Control) Systems Periodic System Physical count of items made at periodic intervals; order is placed for a variable amount after fixed passage of time. Perpetual (Continuous) Inventory System System that keeps track of removals from inventory continuously, thus monitoring current levels of each item (constant amount is ordered when inventory declines to a predetermined level)

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24 Inventory Models Single-Period Inventory Model One time purchasing decision (Examples: selling t- shirts at a football game, newspapers, fresh bakery products, fresh flowers) Seeks to balance the costs of inventory over stock and under stock Multi-Period Inventory Models Fixed-Order Quantity Models Event triggered (Example: running out of stock) Fixed-Time Period Models Time triggered (Example: Monthly sales call by sales representative)

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25 Single-Period Inventory Model

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26 Single-Period Inventory Model In a single-period model, items are received in the beginning of a period and sold during the same period. The unsold items are not carried over to the next period. The unsold items may be a total waste, or sold at a reduced price, or returned to the producer at some price less than the original purchase price. The revenue generated by the unsold items is called the salvage value.

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27 Single period model: It is used to handle ordering of perishables (fresh fruits, flowers) and other items with limited useful lives (newspapers, spare parts for specialized equipment). 3 (Newsboy Problem)

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28 Shortage cost (Cost of Understocking) Shortage cost: generally, this cost represents unrealized profit per unit (C u =Revenue per unit – Cost per unit) If a shortage or stockout cost relates to a spare part for a machine, then shortage cost refers to the actual cost of lost production.

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29 Excess cost (Cost of Over Stocking) Excess cost (C e ): difference between purchase cost and salvage value of items left over at the end of a period. If there is a cost associated with disposing of excess items, the salvage cost will be negative.

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30 Single Period Model Given the costs of overestimating/underestimating demand and the probabilities of various demand sizes the goal is to identify the order quantity or stocking level that will minimize the long-run excess (overstock)or shortage costs (understock).

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31 Demand may be discrete or continuous. The demand of computer, newspaper, etc. is usually an integer. Such a demand is discrete. On the other hand, the demand of gasoline is not restricted to integers. Such a demand is continuous. Often, the demand of perishable food items such as fish or meat may also be continuous. Consider an order quantity Q Let p = probability (demand< Q ) = probability of not selling the Q th item. So, (1- p ) = probability of selling the Q th item. Single-Period Models (Demand Distribution)

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32 Expected loss from the Q th item = Expected profit from the Q th item = So, the Q th item should be ordered if Decision Rule (Discrete Demand): –Order maximum quantity Q such that where p = probability (demand< Q ) Single-Period Models (Discrete Demand)

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33 Single-Period Model The service level is the probability that demand will not exceed the stocking level. The service level determines the amount of stocking level to keep.

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34 Optimal Stocking Level (Choosing optimum Stocking level to minimize these costs is similar to balancing a seesaw ) Service Level So Quantity CeCs Balance point Service level = CsCs Cs + Ce Cs = Shortage cost per unit Ce = Excess cost per unit

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35 Service Level Another way to define Service Level is: proportion of cycles in which no stock-out occurs

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36 Service Level Order Cycle Demand Stock-Outs 11800 2 750 3235 45 41400 51800 6200 10 71500 8 900 91600 10 400 Total 1450 55 Since there are two cycles out of ten in which a stockout occurs, service level is 80%. This translates to a 96% fill rate. There are a total of 1,450 units demand and 55 stockouts (which means that 1,395 units of demand are satisfied).

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37 Single Period Model (Demand is represented by a discrete distribution) Unlike the continuous case where the optimal solution is found by determining S o which makes the distribution function equal to the critical ratio c s / (c s + c e ), in the discrete case, the critical ratio takes place between two values of F(S o ) or F(Q) The optimal S o or Q corresponds to the higher value of F( S o ) or F(Q). (Note that, in the discrete case, the distribution function increases by jumps) SEE EXAMPLES 17 & 18 on page 576

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38 Example : Demand for cookies: Demand Probability of Demand 1,800 dozen0.05 2,0000.10 2,2000.20 2,4000.30 2,6000.20 2,8000.10 3,0000,05 Selling price=$0.69, cost=$0.49, salvage value=$0.29 What is the optimal number of cookies to make? c Single-Period Models (Discrete Demand)

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39 Cs= 0.69-0.49=$0.2, Ce= 0.49-0.29=$0.2 Order maximum quantity, Q such that Demand, Q Probability(demand) Probability(demand
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40 Single-Period Models (Continous Demand) Often the demand is continuous. Even when the demand is not continuous, continuous distribution may be used because the discrete distribution may be inconvenient. We shall discuss two distributions: Uniform distribution Normal distribution

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41 Example 2: The J&B Card Shop sells calendars. The once- a-year order for each years calendar arrives in September. The calendars cost $1.50 and J&B sells them for $3 each. At the end of July, J&B reduces the calendar price to $1 and can sell all the surplus calendars at this price. How many calendars should J&B order if the September-to-July demand can be approximated by a. uniform distribution between 150 and 850 Single-Period Models (Continuous Demand)

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42 Overage cost c e = Purchase price - Salvage value =1.5-1=$0.5 Underage cost c s = Selling price - Purchase price =3-1.5=$1.5 Single-Period Models (Continuous Demand)

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43 p =0.75 Now, find the Q so that p = probability(demand< Q ) =0.75 Q * = a + p ( b - a ) =150+0.75(850-150)=675 Area = 850150 Demand Probability Area = Q * Single-Period Models (Continuous Demand)

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44 Example 3: The J&B Card Shop sells calendars. The once- a-year order for each years calendar arrives in September. The calendars cost $1.50 and J&B sells them for $3 each. At the end of July, J&B reduces the calendar price to $1 and can sell all the surplus calendars at this price. How many calendars should J&B order if the September-to-July demand can be approximated by b. normal distribution with = 500 and =120. Single-Period Models (Continuous Demand)

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45 Solution to Example 3: c e =$0.50, c u =$1.50 (see Example 2) p = = 0.75 Single-Period Models (Continuous Demand)

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46 Single-Period Models (Continuous Demand) Now, find the Q so that p = 0.75

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47 Single-Period Models (Continuous Demand Q= S o = mean + zσ = 500 +.68(120) = 582

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48 Single Period Example 15 (pg. 574) Demand is uniformly distributed Ce = $0.20 per unit Cs = $0.60 per unit Service level = Cs/(Cs+Ce) =.6/(.6+.2) Service level =.75 Opt. Stock.Level=S 0 =300+.75(500-300)= 450 liters Service Level = 75% Quantity CeCs Stockout risk = 1.00 – 0.75 = 0.25

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49 Uniform Distribution [Continuous Distn] A random variable X is uniformly distributed on the interval (a,b), U(a,b), if its pdf and cdf are: Properties –P(x 1 < X < x 2 ) is proportional to the length of the interval [F(x 2 ) – F(x 1 ) = (x 2 -x 1 )/(b-a)] –E(X) = (a+b)/2V(X) = (b-a) 2 /12 U(0,1) provides the means to generate random numbers, from which random variates can be generated.

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50 Poisson Distribution [Discrete Distn] Poisson distribution describes many random processes quite well and is mathematically quite simple. –where > 0, pdf and cdf are: –E(X) = = V(X)

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51 Normal Distribution [Continuous Distn] A normally distributed random variable X has the pdf: –Mean: –Variance: –Denoted as X ~ N(, 2 ) Special properties:. –symmetric about. –The maximum value of the pdf occurs at x = ; the mean and mode are equal.

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52 Normal Distribution [Continuous Distn] Evaluating the distribution: –Use numerical methods (no closed form) –Independent of and using the standard normal distribution: Z ~ N(0,1) –Transformation of variables: let Z = (X - ) /,

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53 Normal Distribution [Continuous Distn] Example: The time required to load an oceangoing vessel, X, is distributed as N(12,4) –The probability that the vessel is loaded in less than 10 hours: Using the symmetry property, (1) is the complement of (-1)

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54 Single Period Model (Demand is represented by a continous distribution) Our college basketball team is playing in a tournament game this weekend. Based on our past experience we sell on average 2,400 shirts with a standard deviation of 350 and we can assume that demand for shirts is approximately normally distributed. We make $10 on every shirt we sell at the game, but lose $5 on every shirt not sold. What is the optimal stocking level for shirts? S o =mean + zσ C s = $10 and C e = $5; P $10 / ($10 + $5) =.667 Z.667 =.432 therefore we need 2,400 +.432(350) = 2,551 shirts

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55 Multi-Period Inventory Models Fixed-Order Quantity Models (Types of) Economic Order Quantity Model (EOQ) Economic Production Order Quantity (Economic Lot Size) Model (EPQ) Economic Order Quantity Model with Quantity Discounts Fixed Time Period (Fixed Order Interval) Models

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56 Fixed Order Quantity Models: Economic Order Quantity Model

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57 Economic Order Quantity Model Assumptions (1 of 2): Demand for the product is known with certainty, it is constant and uniform throughout the period Lead time (time from ordering to receipt) is known and constant Price per unit of product is constant (no quantity discounts). So it is not included in the total cost. Inventory holding cost is based on average inventory

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58 Economic Order Quantity Model Assumptions (2 of 2): Ordering or setup costs are constant All demands for the product will be satisfied (no backorders are allowed) No stockouts (shortages) are allowed The order quantity is received all at once. (Instantaneous receipt of material in a single lot) The goal is to calculate the order quantitiy that minimizes total cost

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59 Basic Fixed-Order Quantity Model and Reorder Point Behavior R = Reorder point Q = Economic order quantity L = Lead time L L QQQ R Time Number of units on hand (Inv. Level) 1. You receive an order quantity Q. 2. You start using them up over time. 3. When you reach down to a level of inventory of R, you place your next Q sized order. 4. The cycle then repeats.

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60 EOQ Model Reorder Point (ROP) Time Inventory Level Average Inventory (Q/2) Lead Time Order Quantity (Q) Demand rate Order placedOrder received

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61 EOQ Cost Model: How Much to Order? By adding the holding and ordering costs together, we determine the total cost curve, which in turn is used to find the optimal order quantity that minimizes total costs Slope = 0 Total Cost Order Quantity, Q Annual cost ($) Minimum total cost Optimal order Q opt Q opt Carrying Cost = HQHQ22HQHQ222 Ordering Cost = SDSDQQSDSDQQQ

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62 More units must be stored if more are ordered Purchase Order DescriptionQty. Microwave1 Order quantity Purchase Order DescriptionQty. Microwave1000 Order quantity Why Holding Costs Increase?

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63 Cost is spread over more units Example: You need 1000 microwave ovens Purchase Order DescriptionQty. Microwave1 Purchase Order DescriptionQty. Microwave1 Purchase Order DescriptionQty. Microwave1 Purchase Order Description Qty. Microwave 1 1000 Order (Postage $ 0.33)1 Order (Postage $330) Order quantity Purchase Order Description Qty. Microwave1000 Why Ordering Costs Decrease ?

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64 Basic Fixed-Order Quantity (EOQ) Model Formula Total Annual = Cost Annual Purchase Cost Annual Ordering Cost Annual Holding Cost ++ TC=Total annual cost D =Annual demand C =Cost per unit Q =Order quantity S =Cost of placing an order or setup cost R =Reorder point L =Lead time H=Annual holding and storage cost per unit of inventory TC=Total annual cost D =Annual demand C =Cost per unit Q =Order quantity S =Cost of placing an order or setup cost R =Reorder point L =Lead time H=Annual holding and storage cost per unit of inventory

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65 EOQ Cost Model Annual ordering cost = S D Q Annual carrying cost = HQHQ22HQHQ222 Total cost = + S D Q H Q 2 TC = + S DQS DQ H Q 2 =- + S DQ2S DQ2 H2H2 TC Q 0 =- + S DQ2S DQ2 H2H2 Q opt = 2SD H Deriving Q opt Proving equality of costs at optimal point = S D Q H Q 2 Q 2 = 2S D H Q opt = 2 S D H Using calculus, we take the first derivative of the total cost function with respect to Q, and set the derivative (slope) equal to zero, solving for the optimized (cost minimized) value of Qopt

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66 Deriving the EOQ We also need a reorder point to tell us when to place an order 1) How much to order? 2) When to order?

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67 Optimal Order Quantity Expected Number of Orders Expected Time Between Orders Working Days / Year Working Days / Year == ×× == == = =× Q* DS H N D Q*Q* T N d D ROPdL 2 EOQ Model Equations

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68 EOQ Example 1 (1 of 3) Annual Demand = 1,000 units Days per year considered in average daily demand = 365 Cost to place an order = $10 Holding cost per unit per year = $2.50 Lead time = 7 days Cost per unit = $15 Given the information below, what are the EOQ and reorder point?

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69 EOQ Example 1(2 of 3) In summary, you place an optimal order of 90 units. In the course of using the units to meet demand, place the next order of 90 units when you only have 20 units left.

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70 EOQ Example I(3 of 3) TC min = SDSDQQSDSDQQQ HQHQ22HQHQ222 (10)(1,000)90 (2,5)(90) 2 TC min = $ 111 + $111 = 22 $ Orders per year =D/Q opt =1000/90 =11 orders/year Order cycle time= 365/(D/Qopt) =365/11 =33.1days =33.1days + +

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71 EOQ Example 2(1 of 2) Annual Demand = 10,000 units Days per year considered in average daily demand = 365 Cost to place an order = $10 Holding cost per unit per year = 10% of cost per unit Lead time = 10 days Cost per unit = $15 Determine the economic order quantity and the reorder point given the following… Determine the economic order quantity and the reorder point given the following…

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72 EOQ Example 2(2 of 2) Place an order for 366 units. When in the course of using the inventory you are left with only 274 units, place the next order of 366 units.

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73 EOQ Example 3 H = $0.75 per yardS = $150D = 10,000 yards Q opt = 2 S D H Q opt = 2(150)(10,000)(0.75) Q opt = 2,000 yards TC min = + S D Q H Q 2 TC min = + (150)(10,000) 2,000(0.75)(2,000)2 TC min = $750 + $750 = $1,500 Orders per year =D/Q opt =10,000/2,000 =5 orders/year Order cycle time =311 days/(D/Q opt ) =311/5 =311/5 =62.2 store days

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74 When to Reorder with EOQ Ordering ? Reorder Point – is the level of inventory at which a new order is placed ROP = d. L Safety Stock - Stock that is held in excess of expected demand due to variable demand rate and/or lead time. Service Level - Probability that demand will not exceed supply during lead time (probability that inventory available during the lead time will meet the demand) 1 - Probability of stockout

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75 Reorder Point Example Demand = 10,000 yards/year Store open 311 days/year Daily demand = 10,000 / 311 = 32.154 yards/day Lead time = L = 10 days R = dL = (32.154)(10) = 321.54 yards

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76 Determinants of the Reorder Point The rate of demand The lead time Demand and/or lead time variability Stockout risk (safety stock)

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77 Answer how much & when to order Allow demand and lead time to vary Follows normal distribution Other EOQ assumptions apply Consider service level & safety stock Service level = 1 - Probability of stockout Higher service level means more safety stock More safety stock means higher ROP Probabilistic Models

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78 Safety Stock LT Time Expected demand during lead time Maximum probable demand during lead time ROP Quantity Safety stock Safety stock reduces risk of stockout during lead time

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79 Reorder Point With Variable Demand Reorder point, R Q LT Time LT Inventory level 0

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80 Reorder Point with a Safety Stock Reorder point, R Q LT Time LT Inventory level 0 Safety Stock

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81 Reorder Point With Variable Demand and Constant Lead Time R = dL + z d L where d=average daily demand L=lead time d =the standard deviation of daily demand z=number of standard deviations corresponding to the service level probability z d L=safety stock

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82 Reorder Point for Service Level Probability of meeting demand during lead time = service level Probability of a stockout R Safety stock dL Expected Demand z d L The reorder point based on a normal distribution of LT demand

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83 Reorder Point for Variable Demand (Example) The carpet store wants a reorder point with a 95% service level and a 5% stockout probability d= 30 yards per day, (demand is normally distributed) d = 5 yards per day For a 95% service level, z = 1.65 R= dL + z d L = 30(10) + (1.65)(5)( 10) = 326.1 yards Safety stock= z d L = (1.65)(5)( 10) = 26.1 yards L= 10 days

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84 Shortages and Service Levels It is also important to specify: 1) Expected number of units short per order cycle E(n) =E(z) σ dLT where E(z) is standardized number of units short obtained from Table 12.3, pg. 569. 2) Expected number of units short per year E(N) =E (n) (D/Q) 3) Annual Service Level SL annual = 1- E(N)/D that is percentage of demand filled directly from inventory, known also as FILL RATE.

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85 Example 10 (pg. 568)– shortages and service levels Suppose standard deviation of lead time demand is known to be 20 units. Lead time demand is approximately normal. (a) For lead time service level of 90 percent, determine the expected number of units short for any other cycle. (b) What lead time service level would imply an expected shortage of 2 units?

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86 Answer – shortage and service levels (a) For lead time service level of 90 percent, determine the expected number of units short for any other cycle. σ dLT = 20 units lead time service level is 0.90 from z table (lead time), E(z)= 0.048 (page 569, table 12.3) E(n) =E(z) σ dLT = (0.048) (20)= 0.96 or about 1 unit. (b) What lead time service level would an expected shortage of 2 units imply? E(n) = 2 E(n) =E(z) σ dLT or E(z) = E(n) / σ dLT =(2)/(20)= 0.100 from the table, lead time service level is 81.06 percent or 81.7%

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87 Shortages and Service Levels Expected number of units short per year See example 11, page 568 Annual Service Level See example 12, page 570 Note that annual service level will usually be grater than the cycle service level

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88 Fixed Order Quantity Models: -Noninstantaneous Receipt- Production Order Quantity (Economic Lot Size) Model

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89 Production done in batches or lots Capacity to produce a part exceeds that parts usage or demand rate Allows partial receipt of material Other EOQ assumptions apply Suited for production environment Material produced, used immediately Provides production lot size Lower holding cost than EOQ model Answers how much to order and when to order Production Order Quantity Model

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90 POQ Model Inventory Levels (1 of 2) Inventory Level Time Supply Begins Supply Ends Production portion of cycle Demand portion of cycle with no supply Maximum inventory level

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91 POQ Model Inventory Levels (2 of 2) Time Inventory Level Production Portion of Cycle Max. Inventory Q/p·(p- u) Q* Supply Begins Supply Ends Inventory level with no demand Demand portion of cycle with no supply Average inventory Q/2(1- u/p)

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92 D = Demand per year S = Setup cost H = Holding cost d = Demand per day p = Production per day POQ Model Equations Setup Cost Holding Cost = * = * = Q D Q S ( 1/2 * H * Q - u p 1 ) - u p 1 () Maximum inventory level

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93 Production Order Quantity Example (1 of 2) H = $0.75 per yardS = $150D = 10,000 yards u = 10,000/311 = 32.2 yards per dayp = 150 yards per day POQ opt = = = 2,256.8 yards 2 S D H 1 - upup 2(150)(10,000) - 0.75 1 - 32.2 150 TC = + 1 - = $1,329 upup S D Q H Q 2 Production run = = = 15.05 days per order QpQp 2,256.8 150

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94 Production Quantity Example (2 of 2) H = $0.75 per yardS = $150D = 10,000 yards u= 10,000/311 = 32.2 yards per dayp = 150 yards per day Q opt = = = 2,256.8 yards 2C o D C c 1 - dp 2(150)(10,000) 0.75 1 - 32.2150 TC = + 1 - = $1,329 dp CoDCoDQQCoDCoDQQQ CcQCcQ22CcQCcQ222 Production run = = = 15.05 days per order QpQp 2,256.8 150 Number of production runs = = = 4.43 runs/year DQDQ 10,000 2,256.8 Maximum inventory level =Q 1 - = 2,256.8 1 - =1,772 yards upup 32.2 150

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95 Fixed-Order Quantity Models: Economic Order Quantity Model with Quantity Discounts

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96 Answers how much to order & when to order Allows quantity discounts –Price per unit decreases as order quantity increases –Other EOQ assumptions apply Trade-off is between lower price & increased holding cost Quantity Discount Model TC = + + PD S D Q Q iP Q2 Where P: Unit Price Total cost with purchasing cost

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97 Total Costs with Purchasing Cost Cost EOQ TC with PD TC without PD PD 0 Quantity Adding Purchasing cost doesnt change EOQ

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98 Quantity Discount Models There are two general cases of quantity discount models: 1.Carrying costs are constant (e.g. $2 per unit). 2.Carrying costs are stated as a percentage off purchase price (20% of unit price)

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99 1) Total Cost with Constant Carrying Costs (Compute the Common Optimal Order Quantity OC EOQ Quantity Total Cost TC a TC c TC b Decreasing Price CC a,b,c

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100 2) Total Cost with Variable Carrying Cost (Compute Optimal Order Quantity for each price range) Based on the same assumptions as the EOQ model, the price-break model has a similar Q opt formula: i = percentage of unit cost attributed to carrying inventory C = cost per unit Since C changes for each price-break, the formula above will have to be used with each price-break cost value

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101 Quantity Discount – How Much to Order?

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102 Price-Break Example 1 (1 of 3) ORDER SIZEPRICE 0 - 99 $10 100 - 1998 ( d 1 ) 200+6 ( d 2 ) For this problem holding cost is given as a constant value, not as a percentage of price, so the optimal order quantity is the same for each of the price ranges. (see the figure 12.9)

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103 Price Break Example 1 (2 of 3) Q opt Carrying cost Ordering cost Inventory cost ($) Q( d 1 ) = 100 Q( d 2 ) = 200 TC ( d 2 = $6 ) TC ( d 1 = $8 ) TC = ($10 )

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104 Price Break Example 1 (3 of 3) Q opt Carrying cost Ordering cost Inventory cost ($) Q( d 1 ) = 100 Q( d 2 ) = 200 TC ( d 2 = $6 ) TC ( d 1 = $8 ) TC = ($10 ) The lowest total cost is at the second price break

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105 Price Break Example 2 QUANTITYPRICE 1 - 49$1,400 50 - 891,100 90+900 S =$2,500 S =$2,500 H =$190 per computer D =200 Q opt = = = 72.5 PCs 2SD2SDHH2SD2SDHHH2(2500)(200)190 TC = + + PD = $233,784 SD Q opt H Q opt 2 For Q = 72.5 TC = + + PD = $194,105 SDSDQQSDSDQQQ H Q 2 For Q = 90

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106 Price-Break Example 3 (1 of 4) A company has a chance to reduce their inventory ordering costs by placing larger quantity orders using the price-break order quantity schedule below. What should their optimal order quantity be if this company purchases this single inventory item with an e-mail ordering cost of $4, a carrying cost with a rate of 2% of the unit price, and an annual demand of 10,000 units? Order Quantity(units)Price/unit($) 0 to 2,499 $1.20 2,500 to 3,999 1.00 4,000 or more.98

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107 Price-Break Example (2 of 4) Annual Demand (D)= 10,000 units Cost to place an order (S)= $4 First, plug data into formula for each price-break value of C Carrying cost % of total cost (i)= 2% Cost per unit (C) = $1.20, $1.00, $0.98 Interval from 0 to 2499, the Q opt value is feasible Interval from 2500-3999, the Q opt value is not feasible Interval from 4000 & more, the Q opt value is not feasible Next, determine if the computed Q opt values are feasible or not

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108 Price-Break Example 2 (3 of 4) Since the feasible solution occurred in the first price- break, it means that all the other true Q opt values occur at the beginnings of each price-break interval. Why? 0 1826 2500 4000 Order Quantity Total annual costs So the candidates for the price- breaks are 1826, 2500, and 4000 units Because the total annual cost function is a u shaped function

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109 Price-Break Example 2 (4 of 4) Next, we plug the true Q opt values into the total cost annual cost function to determine the total cost under each price-break TC(0-2499)=(10000*1.20)+(10000/1826)*4+(1826/2)(0.02*1.20) = $12,043.82 TC(2500-3999)= $10,041 TC(4000&more)= $9,949.20 TC(0-2499)=(10000*1.20)+(10000/1826)*4+(1826/2)(0.02*1.20) = $12,043.82 TC(2500-3999)= $10,041 TC(4000&more)= $9,949.20 Finally, we select the least costly Q opt, which in this problem occurs in the 4000 & more interval. In summary, our optimal order quantity is 4000 units

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110 Multi-period Inventory Models: Fixed Time Period (Fixed-Order- Interval) Models

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111 Orders are placed at fixed time intervals Order quantity for next interval? (inventory is brought up to target amount, amount ordered varies) Suppliers might encourage fixed intervals Requires only periodic checks of inventory levels (no continous monitoring is required) Risk of stockout between intervals Fixed-Order-Interval Model

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112 Inventory Level in a Fixed Period System Various amounts (Q i ) are ordered at regular time intervals (p) based on the quantity necessary to bring inventory up to target maximum ppp Q1Q1Q1Q1 Q2Q2Q2Q2 Q3Q3Q3Q3 Q4Q4Q4Q4 Target maximum Time d Inventory

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113 Tight control of inventory items Items from same supplier may yield savings in: Ordering Packing Shipping costs May be practical when inventories cannot be closely monitored Fixed-Interval Benefits

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114 Requires a larger safety stock Increases carrying cost Costs of periodic reviews Fixed-Interval Disadvantages

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115 Fixed-Time Period Model with Safety Stock Formula q = Average demand + Safety stock – Inventory currently on hand

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116 Fixed-Time Period Model: Determining the Value of T+L The standard deviation of a sequence of random events equals the square root of the sum of the variances

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117 Order Quantity for a Periodic Inventory System Q = d(t b + L) + z d T + L - I where d= average demand rate T= the fixed time between orders L= lead time d = standard deviation of demand z d T + L= safety stock I= inventory level z = the number of standard deviations for a specified service level

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118 Fixed-Period Model with Variable Demand (Example 1) d= 6 bottles per day d = 1.2 bottles T= 60 days L= 5 days I= 8 bottles z= 1.65 (for a 95% service level) Q= d(T + L) + z d T + L - I = (6)(60 + 5) + (1.65)(1.2) 60 + 5 - 8 = 397.96 bottles

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119 Fixed-Time Period Model with Variable Demand (Example 2)(1 of 3) Average daily demand for a product is 20 units. The review period is 30 days, and lead time is 10 days. Management has set a policy of satisfying 96 percent of demand from items in stock. At the beginning of the review period there are 200 units in inventory. The standard deviation of daily demand is 4 units. Given the information below, how many units should be ordered?

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120 Fixed-Time Period Model with Variable Demand (Example 2)(2 of 3) So, by looking at the value from the Table, we have a probability of 0.9599, which is given by a z = 1.75

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121 Fixed-Time Period Model with Variable Demand (Example 2) (3 of 3) So, to satisfy 96 percent of the demand, you should place an order of 645 units at this review period

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122 ABC Classification System Demand volume and value of items vary Items kept in inventory are not of equal importance in terms of: – dollars invested – profit potential – sales or usage volume – stock-out penalties

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123 ABC Classification System Classifying inventory according to some measure of importance and allocating control efforts accordingly. A A - very important B B - mod. important C C - least important Annual $ value of items A B C High Low High Percentage of Items

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124 Classify inventory into 3 categories typically on the basis of the dollar value to the firm $ volume = Annual demand x Unit cost A class, B class, C class Policies based on ABC analysis –Develop class A suppliers more carefully –Give tighter physical control of A items –Forecast A items more carefully ABC Analysis

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125 % of Inventory Items Classifying Items as ABC 0 20 40 60 80 100 050100 % Annual $ UsageA B C Class% $ Vol% Items A70-805-15 B1530 C 5-1050-60

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126 ABC Classification 1$ 6090 235040 330130 48060 530100 620180 710170 832050 951060 1020120 PARTUNIT COSTANNUAL USAGE

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127 ABC Classification 1$ 6090 235040 330130 48060 530100 620180 710170 832050 951060 1020120 PARTUNIT COSTANNUAL USAGE TOTAL% OF TOTAL% OF TOTAL PARTVALUEVALUEQUANTITY% CUMMULATIVE 9$30,60035.96.06.0 816,00018.75.011.0 214,00016.44.015.0 15,4006.39.024.0 44,8005.66.030.0 33,9004.610.040.0 63,6004.218.058.0 53,0003.513.071.0 102,4002.812.083.0 71,7002.017.0100.0 $85,400

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128 ABC Classification 1$ 6090 235040 330130 48060 530100 620180 710170 832050 951060 1020120 PARTUNIT COSTANNUAL USAGE TOTAL% OF TOTAL% OF TOTAL PARTVALUEVALUEQUANTITY% CUMMULATIVE 9$30,60035.96.06.0 816,00018.75.011.0 214,00016.44.015.0 15,4006.39.024.0 44,8005.66.030.0 33,9004.610.040.0 63,6004.218.058.0 53,0003.513.071.0 102,4002.812.083.0 71,7002.017.0100.0 $85,400 A B C

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129 ABC Classification 1$ 6090 235040 330130 48060 530100 620180 710170 832050 951060 1020120 PARTUNIT COSTANNUAL USAGE TOTAL% OF TOTAL% OF TOTAL PARTVALUEVALUEQUANTITY% CUMMULATIVE 9$30,60035.96.06.0 816,00018.75.011.0 214,00016.44.015.0 15,4006.39.024.0 44,8005.66.030.0 33,9004.610.040.0 63,6004.218.058.0 53,0003.513.071.0 102,4002.812.083.0 71,7002.017.0100.0 $85,400 A B C % OF TOTAL CLASSITEMSVALUEQUANTITY A9, 8, 271.015.0 B1, 4, 316.525.0 C6, 5, 10, 712.560.0

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130 ABC Classification 100 100 – 80 80 – 60 60 – 40 40 – 20 20 – 0 0 – |||||| 020406080100 % of Quantity % of Value A B C

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131 Inventory accuracy refers to how well the inventory records agree with physical count. Cycle counting refers to Physical Count of items in inventory. Used often with ABC classification –While A items are counted most often (e.g., daily), C items are counted the least frequently. Inventory Accuracy and Cycle Counting

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132 Last Words Inventories have certain functions. But too much inventory -Tends to hide problems -Costly to maintain So it is desired Reduce lot sizes Reduce safety stocks

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